A094028 Expansion of 1/((1-x)*(1-100*x)).

1, 101, 10101, 1010101, 101010101, 10101010101, 1010101010101, 101010101010101, 10101010101010101, 1010101010101010101, 101010101010101010101, 10101010101010101010101, 1010101010101010101010101, 101010101010101010101010101, 10101010101010101010101010101

Offset: 0

Author

Paul Barry, Apr 22 2004

Keywords

• easy
• nonn

Comments

Regarded as binary numbers and converted to decimal, these become 1,5,21,85,... the partial sums of 4^n (see A002450).

Partial sums of 100^n.

Odd terms of A056830. - Alexandre Wajnberg, May 31 2005

101 is the only term that is prime, since (100^k-1)/99 = (10^k+1)/11 * (10^k-1)/9. When k is odd and not 1, (10^k+1)/11 is an integer > 1 and thus (100^k-1)/99 is nonprime. When k is even and greater than 2, (100^k-1)/99 has the prime factor 101 and is nonprime. - Felix Fröhlich, Oct 17 2015

Previous comment is the answer to the problem A1 proposed during the 50th Putnam Competition in 1989 (link). - Bernard Schott, Mar 24 2023

Examples

			From _Omar E. Pol_, Dec 13 2008: (Start)
=======================
n ....... a(n)
0 ........ 1
1 ....... 101
2 ...... 10101
3 ..... 1010101
4 .... 101010101
5 ... 10101010101
======================
(End)
		

References

• Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
• Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Links

• Robert Price, Table of n, a(n) for n = 0..999
• Kiran S. Kedlaya, The 50th William Lowell Putnam Mathematical Competition, Problem A1, Dec 02 1989.
• J. V. Leyendekkers and A.G. Shannon, Modular Rings and the Integer 3, Notes on Number Theory & Discrete Mathematics, 17 (2011), pp. 47-51.
• Robert Price, Comments on A094028 concerning Elementary Cellular Automata, Feb 21 2016.
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
• Stephen Wolfram, A New Kind of Science.
• Index to Elementary Cellular Automata.
• Index entries for sequences related to cellular automata.
• Index entries for linear recurrences with constant coefficients, signature (101,-100).
• Index to sequences related to Olympiads and other Mathematical Competitions.

Crossrefs

Bisection of A147759. [Omar E. Pol, Nov 13 2008]

Cf. A002450, A056830, A094027.

Cf. similar sequences of the form (k^n-1)/(k-1) listed in A269025.

Programs

Magma

[1+100*(100^n-1)/99 : n in [0..15]]; // Wesley Ivan Hurt, Oct 17 2015

Maple

A094028:=n->1+100*(100^n-1)/99: seq(A094028(n), n=0..15); # Wesley Ivan Hurt, Oct 17 2015

Mathematica

CoefficientList[Series[1/((1-x)(1-100x)),{x,0,20}],x] (* or *) Table[ FromDigits[ PadRight[{},2n-1,{1,0}]],{n,20}] (* or *) LinearRecurrence[ {101,-100},{1,101},20] (* or *) NestList[100#+1&,1,20] (* Harvey P. Dale, Apr 27 2015 *)

Maxima

A094028(n):=1+100*(100^n-1)/99$
makelist(A094028(n),n,0,30); /* Martin Ettl, Nov 06 2012 */

PARI

a(n) = 1+100*(100^n-1)/99 \\ Felix Fröhlich, Oct 17 2015

PARI

Vec(1/((1-x)*(1-100*x)) + O(x^100)) \\ Altug Alkan, Oct 17 2015

Formula

G.f.: 1/((1-x)*(1-100*x)).

a(n) = 1 + 100*(100^n-1)/99. - N. J. A. Sloane, Apr 20 2008

a(n) = 100^(n+1)/99 - 1/99.

a(n) = A094027(2*n+1).

a(n) = 100*a(n-1) + 1, a(0) = 1. - Philippe Deléham, Feb 22 2014

a(n) = 101*a(n-1) - 100*a(n-2) for n > 1. - Wesley Ivan Hurt, Oct 17 2015

a(n) = (100^(n+1) - 1)/99. - Bernard Schott, Apr 15 2021

E.g.f.: exp(x)*(100*exp(99*x) - 1)/99. - Elmo R. Oliveira, Mar 06 2025